Queueing
Overview
A queueing system is a system where individuals enter the system at random intervals,
at which point, the individual spends an amount of time in the system, at which point the
individual leaves the system. An example of a Queueing system would be
customers calling a help line, or citizens entering a line at the DMV.
Modeling
A simple queueing model can be defined as
- {% A^k %} be the the arrive time of customer k
- {% W^k %} be the wait time of customer k before exiting the
system.
- {% A(t) %} be the cumulative number of arrivals into the system by time t.
- {% N(t) %} is the number of customers in the system
Rate Definitions
The long run rate of arrivals is defined to be
{% \lambda = lim_{t \rightarrow \infty } \frac{A(t)}{t} %}
The average time spent in the system.
{% W = lim_{k \rightarrow \infty} \frac{1}{k} \sum_{i=1}^k W^k %}
The average number of individuals in the system.
{% N = lim_{T \rightarrow \infty} \frac{1}{T} \int_0^T N(t)dt %}
Topics
- Littles Law - relates the average arrival rate, the average time in
the system, and the average number of individuals in the system for a broad spectrum of queueing systems.
